Design and Analysis of Algorithms for Structured Optimization

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This project aims to develop advanced tools for analyzing algorithms that solve structured optimization problems, which play an important role for models in various scientific and engineering fields, including massive data analysis, machine learning, signal processing, and image reconstruction. While there are several practical and successful algorithms for optimizing these frameworks, their fundamental convergence theory is not yet fully understood. This project seeks to develop new tools that will enable a better understanding of the core features of both the models and algorithms, design more effective algorithms, and tackle more challenging applications. The outcomes of this project will contribute to a better understanding of how to achieve fast convergence in modified classical iterations, which will improve their efficiency. The project will integrate its findings into graduate-level courses and engage Ph.D. students in research related to the project's topics.This research project will focus on designing and analyzing novel efficient projection/proximal-type schemes for solving (non)convex and (non)smooth composite optimization and feasibility problems. The research will investigate how the irregular phenomena of nonsmoothness and nonconvexity affect algorithmic performance and will study the possibility of improving the convergence complexity of the algorithms by exploiting the particular structure of the problem. Splitting iterations frequently show signs of zigzagging, affecting those schemes' convergence speed. The proposed research will advance and adapt the Circumcentered-Reflection Method to enhance the performance and complexity of splitting algorithms for solving more general structured problems. In the absence of classical assumptions, the project will also investigate variations of the FISTA algorithm for solving composite problems and semismooth Newtonian iterations for solving generalized projection equations and complementarity problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.